x_{1}(n) -> y_{1}(n) | => ax_{1}(n) + bx_{2}(n) -> ay_{1}(n) + by_{2}(n) |
x_{2}(n) -> y_{2}(n) | |
The discrete unit impulse function is defined by
(n) = | 1, | n = 0 |
0, | n 0. | |
It can be shown that the output is
which is a discrete convolution y(n) = h(n) * x(n). As in the analog case, now instead of the Fourier transform we have the Z-transform, we obtain
H(z) = | Y(z) | . |
X(z) | ||
H(z) is called the transfer function of the system. It is the Z-transform of the impulse response h(n),
H(z) = h(k) z^{-k} |
The Z-transforms X(z) and Y(z) are defined similarly.
Example 3.2.1: Finding the transfer function
A discrete system is causal if h(n) = 0, when n < 0. Then
If also the input is causal, x(n) = 0 when n < 0, then
Example 3.2.2: Digital derivator